This is the eleventh in a series of positive self contained postings to fom
covering a wide range of topics in f.o.m. Previous ones are:
1:Foundational Completeness 11/3/97, 10:13AM, 10:26AM.
2:Axioms 11/6/97.
3:Simplicity 11/14/97 10:10AM.
4:Simplicity 11/14/97 4:25PM
5:Constructions 11/15/97 5:24PM
6:Undefinability/Nonstandard Models 11/16/97 12:04AM
7.Undefinability/Nonstandard Models 11/17/97 12:31AM
8.Schemes 11/17/97 12:30AM
9:Nonstandard Arithmetic 11/18/97 11:53AM
10:Pathology 12/8/97 12:37AM
A complete archiving of fom, message by message, is available at
http://www.math.psu.edu/simpson/fom/index.html
Also, my series of positive postings (only) is archived at
http://www.math.ohio-state.edu/foundations/manuscripts.html
NOTE: From now on, I will use FOM for Foundations of Mathematics, and fom
for the mailing list. I am tired of putting the periods in.
FOM is not appropriately viewed as a branch of mathematics. It's methods
are highly mathematical. Obviously its subject matter is appropriately
mathematical because it is Foundations of Mathematics. Since it is not a
branch of mathematics, it is inappropriate to judge it as a branch of
mathematics - like differential geometry or algebraic number theory.
The same kind of thing is true of, say, statistics. Its methods are highly
mathematical. Its subject matter is appropriately mathematical given its
quantitative aims. It is not a branch of mathematics, and it is
inappropriate to judge it as a branch of mathematics - like differential
geometry or algebraic number theory.
So if FOM is not a branch of mathematics, then what is it? Before
addressing this question in specific detail (in later postings), let me
comment on a related issue that may serve as a useful preliminary: what is
mathematical logic, and how is it related to FOM?
The great successes in FOM starting with Frege, through Godel, and up to
the present, required technical developments that were either done on the
spot in an essentially self contained way, or used earlier technical
developments. The earlier successes were of course more self contained.
Also these great successes involved key definitions of various mathematical
structures. These spawned many new mathematical subjects, which were
systematically pursued. As is usual in mathematics, these systematic
investigations attain a life of their own, independently of their origins
(from FOM). This is inevitable and natural. One desires to have an arsenal
of techniques and results concerning the great definitions of FOM.
Experience in FOM has shown that without a certain amount of systematic
development of subjects spawned by the great successes of FOM, progress in
FOM would be practically impossible. A lot of modern FOM requires
systematic consideration of hundreds, or thousands, of novel combinations
of ideas and approaches. Almost none of these have any permanent value. To
find the ones of permanent value, one must reject hundreds, or thousands,
of related combinations and approaches. This is simply not possible to do
in any kind of real time without the benefit of a big arsenal of
techniques, results, and intuitions arising from the development of
mathematical logic.
Thus the distinction between mathematical logic and FOM is this: the latter
comprises the development of subjects directly arising out of the
spectacular work in FOM - but without regard to the relationship of this
development to FOM. The development proceeds substantially in the way that
many areas of mathematics proceeds - where one seeks to sort out basic
relationships, isolate important cases, classify objects that arise, make
additional definitions, etcetera. The usual evolutionary process takes
hold, where certain problems remain unsolved, and special recognition
awaits the solver.
This development of mathematical logic proceeds largely independently of
its connection with FOM. Similarly, the development of most areas of
mathematics also proceeds largely independently of its connections with
wider issues. In some parts of mathematical logic, there is some
consideration paid to FOM; in other parts, virtually none. The same with
most areas of mathematics.
However, this disconnect with FOM creates a special problem for
mathematical logic. Since FOM is no longer the motivating force behind most
of mathematical logic, there is the real problem of how to evaluate it.
Virtually all of research in mathematical logic is now housed in
mathematics departments, and so it is natural for this research to be
evaluated as a branch of mathematics. Mathematical logicians cannot, in
general, cast the importance of what they do in terms of FOM. So there is a
question of how mathematical logicians can relate their work to the
mathematical community.
Only a handful of mathematical logicians have been able to solve this
problem, and their solutions have pretty much been one of the following:
i) move into direct applications to mathematics;
ii) move into applications to computer science, or into computer science
itself.
Since the subject matter of computer science is very fluid, the borderline
between applications of logic to computer science and computer science
itself is murky. Often ii) involves moving to a computer science department.
I want to confine myself to i). The standard branches of mathematical logic
are usually considered to be (in alphabetical order): model theory, proof
theory, recursion theory, set theory. This move is far more pronounced and
successful in model theory than in any other of the standard branches. It
is also, secondarily, somewhat successful in descriptive set theory.
Only time will tell how the recent successes in applications of model
theory to mathematics are received by mathematicians, and what effect this
will have on the status of model theorists who have various degrees of
involvement in these recent successes.
Many of the leading people in applied model theory are subscribers to the
fom, and some time ago, there was a lot of feverish postings concerning the
meaning of FOM. Many of these applied model theorists sought to cast some
of their work as FOM, and I was one of the people who most ardently
resisted this characterization. They have shown that notions, structures,
and methods from model theory can bear fruitfully on some situations in
algebraic number theory and related areas to a greater extent than had been
previously thought by everybody. This carries on a tradition of Tarski, A.
Robinson, James Ax, Simon Kochen, and others.
As I will again make clear when I get into detail about what FOM is,
applied model theory is naturally and appropriately distinguished from FOM.
But this is not to say that applied model theorists themselves couldn't
contribute to basic issues in FOM that are related to some of what they do.
For instance, there is the matter of "what is a well behaved mathematical
structure?" that could be given a suitably gripping answer that would be
regarded as a work in FOM. It could have the appropriate kind of general
intellectual significance.
But what of mathematical logic that is not applied nor FOM? This is where
the real difficulties lie. Here are some of my views on the matter.
1. There are many missed opportunities for a redirection of much of this
work towards issues in FOM. For instance, for recursion theory, there is
reverse mathematics, Church's thesis, and also developing decision
procedures for new classes of mathematical statements. Of course, the
latter may equally well be classified as model theory. In model theory,
there is "what is a well behaved mathematical structure?" For set theory,
there is alternative approaches to set theoretical axioms other than
through realism and Platonism, and also more deeply analyzing ZFC and its
fragments, and small large cardinals, and getting a general handle on what
kinds of set theoretic problems are independent and what kinds are not. For
proof theory, there is the matter of analyzing the structure of actual
mathematical proofs, and identifying new significant features of them. This
is just a small sample. I hope to devote postive postings to all of these
and more.
2. Some of set theory and proof theory undeniably is motiviated by some
outlooks on FOM. But these outlooks are relatively undeveloped and have not
been seriously reexamined for decades. I'm thinking of the realism and
Platonism behind much of current work in set theory, and the idea of
"finitist consistency proofs" by ever more complicated ordinal notation
systems in proof theory. The limitations and drawbacks of these approaches
are pretty much recognized by everyone involved and not involved, but in
the current atmosphere of disconnect between mathematical logic and FOM,
these limitations and drawbacks are de-emphasized. In this atmosphere, the
feeling is that if it leads to complicated and intricate work, then it is
OK - despite limitations and drawbacks. I, for one, have always called for
a perpetual rethinking of the underlying assumptions behind these, or any
technical developments. If there is a better motivating idea that would
push the technical development in an altered direction, then so be it. If
that means abandoning longstanding technical projects, then fine. If people
persist in resisting the inevitable, then others will come in and steal
their thunder.
3. More bluntly - mathematics, including mathematical logic, operates on a
kind of code of silence. One simply doesn't want to talk openly about
significance; particularly about other ways of looking at things that may
assume greater significance, and involve a change in research perspective.
One lapses into: well, if its hard, complicated, and intricate, and made
some sense some time, then it is OK; and it is OK to judge everybody's work
in these terms - i.e., is it hard, complicated, and intricate? How hard,
complicated, and intricate?
4. But whereas 3 is a time honored way that most fields of mathematics
operate, I don't think that it can really work for the mathematical logic
that is disconnected from applications and from FOM. It is in danger of
being marginalized.
Next time: focus on Foundations and FOM.